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DIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS

Published online by Cambridge University Press:  02 February 2004

GWYNETH M. STALLARD
GWYNETH M. STALLARD
Affiliation:
Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA g.m.stallard@open.ac.uk
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Abstract

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in {\mathbb N}$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.

Keywords

30D0537F1037F1537F3537F50
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 36 , Issue 2 , March 2004 , pp. 263 - 270
Copyright
© The London Mathematical Society 2004

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